Topological and Symbolic Dynamics for Axiom A Diffeomorphisms with Holes

We consider an Axiom A diffeomorphism and the invariant set $\Omega$ of orbits which never falls into a fixed hole. We study various aspects of the complexity of the symbolic representation of $\Omega$. Our main result are that each topologically transitive component of $\Omega$ is coded and that typically $\Omega$ is of finite type.